Quotient Rule Differentiation Examples for High School Calculus

📚 Quick Study Guide

🔍 The Quotient Rule helps you differentiate functions that are expressed as a ratio (fraction) of two other functions. 💡 The formula for the quotient rule is: If $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$. 📝 Remember that $u$ is the numerator and $v$ is the denominator. ➕ Always be careful with the order of terms in the numerator; subtraction is not commutative. ➗ The denominator is $v^2$, i.e., the square of the original denominator.

🧪 Practice Quiz

1. What is the derivative of $y = \frac{x^2}{x+1}$?
1. $\frac{x^2 + 2x}{(x+1)^2}$
2. $\frac{2x}{1}$
3. $\frac{x^2 - 2x}{(x+1)^2}$
4. $\frac{x^2 + x}{(x+1)^2}$
2. Find $\frac{dy}{dx}$ if $y = \frac{\sin(x)}{x}$.
1. $\frac{x\cos(x) - \sin(x)}{x^2}$
2. $\frac{\cos(x)}{1}$
3. $\frac{\sin(x) - x\cos(x)}{x^2}$
4. $\frac{-\cos(x)}{x^2}$
3. Differentiate $f(x) = \frac{e^x}{x^2}$.
1. $\frac{e^x(x^2 - 2x)}{x^4}$
2. $\frac{e^x}{2x}$
3. $\frac{e^x(x^2 + 2x)}{x^4}$
4. $\frac{e^x}{x^4}$
4. What is the derivative of $y = \frac{x+2}{x-2}$?
1. $\frac{4}{(x-2)^2}$
2. $\frac{2x}{(x-2)^2}$
3. $\frac{-4}{(x-2)^2}$
4. $\frac{0}{(x-2)^2}$
5. If $y = \frac{x}{\cos(x)}$, find $\frac{dy}{dx}$.
1. $\frac{\cos(x) + x\sin(x)}{\cos^2(x)}$
2. $\frac{1}{-\sin(x)}$
3. $\frac{\cos(x) - x\sin(x)}{\cos^2(x)}$
4. $\frac{-\sin(x)}{\cos^2(x)}$
6. Differentiate $f(x) = \frac{\ln(x)}{x}$.
1. $\frac{1 - \ln(x)}{x^2}$
2. $\frac{1}{x^2}$
3. $\frac{\ln(x) - 1}{x^2}$
4. $\frac{1}{x}$
7. Find the derivative of $y = \frac{x^3}{e^x}$.
1. $\frac{3x^2 - x^3}{e^x}$
2. $\frac{3x^2}{e^x}$
3. $\frac{x^2(3 - x)}{e^x}$
4. $\frac{x^3 - 3x^2}{e^x}$